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Nonemptiness and Compactness of Solution Sets to Generalized Polynomial Complementarity Problems

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Abstract

In this paper, we investigate the generalized polynomial complementarity problem, which is a subclass of generalized complementarity problems with the involved map pairs being two polynomials. Based on the analysis on two structured tensor pairs located in the heading items of polynomials involved, and by using the degree theory, we achieve several results on the nonemptiness and compactness of solution sets. When generalized polynomial complementarity problems reduce to polynomial complementarity problems (or tensor complementarity problems), our results reduce to the existing ones. In particular, one of our results broadens the one proposed in a very recent paper to guarantee the nonemptiness and compactness of solution sets to generalized polynomial complementarity problems. Furthermore, we establish several existence and uniqueness results, which enrich the theory of generalized complementarity problems with the observation that some known conditions to guarantee the existence and uniqueness of solutions may not hold for a lot of generalized polynomial complementarity problems.

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Acknowledgements

The authors are very thankful to the anonymous reviewers for their useful comments and constructive advice. The second author’s work is partially supported by the National Natural Science Foundation of China (Grant No. 11871051).

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Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Meng-Meng Zheng, Zheng-Hai Huang & Xiao-Xiao Ma

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  1. Meng-Meng Zheng
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  2. Zheng-Hai Huang
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  3. Xiao-Xiao Ma
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Correspondence to Zheng-Hai Huang.

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Communicated by Guoyin Li.

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Zheng, MM., Huang, ZH. & Ma, XX. Nonemptiness and Compactness of Solution Sets to Generalized Polynomial Complementarity Problems. J Optim Theory Appl 185, 80–98 (2020). https://doi.org/10.1007/s10957-020-01645-6

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  • DOI: https://doi.org/10.1007/s10957-020-01645-6

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